binomialsexpressions

Binomial expressions are polynomials that contain exactly two terms, connected by addition or subtraction. Each term is a monomial with a nonzero coefficient. Examples include x^2 + 3x and 7y − 4. If like terms appear, they can be combined to simplify the expression, potentially changing the number of terms.

When a binomial is raised to a power, its expansion is described by the binomial theorem: (u + v)^n = sum from k = 0 to n of C(n, k) u^(n−k) v^k, where C(n, k) is the binomial coefficient. The coefficients form Pascal’s triangle. For instance, (x + 1)^3 = x^3 + 3x^2 + 3x + 1.

Binomial expressions can often be factored or simplified. If the two terms share a common factor, factor it out: for example 6x^3 + 9x^2 = 3x^2(2x + 3). Some binomials exhibit special products, such as a^2 − b^2 = (a − b)(a + b).

Binomial expressions are a fundamental concept in algebra and appear in various areas, including polynomial identities, factoring, and solving equations. They also relate to probability through the binomial distribution, where binomial coefficients count combinations of successes in a fixed number of trials. The term binomial refers to the two terms that comprise the expression.

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